Jet bundle

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In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form.

Historically, jet bundles are attributed to Ehresmann, and were an advance on the method (prolongation) of Elie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly-introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.)

More recently, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach.

Contents

[edit] Jets

Main article: Jet (mathematics).

Let (\mathcal{E}, \pi, \mathcal{M}) be a fiber bundle in a category of manifolds and let p \in \mathcal{M}, with \dim\mathcal{M}=m. Let \Gamma(\pi)\, denote the set of all local sections whose domain contains p\,. Let I=(I(1),I(2),\ldots,I(m)) be a multi-index (an ordered m-tuple of integers), then

|I| := \sum_{i=1}^{m} I(i)
\frac{\partial^{|I|}}{\partial x^{I}} := \prod_{i=1}^{m} \left( \frac{\partial}{\partial x^{i}} \right)^{I(i)}.

Define the local sections \sigma, \eta \in \Gamma(\pi) to have the same r\,-jet at p\, if

\left.\frac{\partial^{|I|} \sigma^{\alpha}}{\partial x^{I}}\right|_{p} = \left.\frac{\partial^{|I|} \eta^{\alpha}}{\partial x^{I}}\right|_{p}, \quad 1 \leq |I| \leq r

The relation that two maps have the same r-jet is an equivalence relation. An r-jet is an equivalence class under this relation, and the r-jet with representative \sigma\, is denoted j^{r}_{p}\sigma. The integer r is also called the order of the jet.

p\, is the source of j^{r}_{p}\sigma.

\sigma\,(p) is the target of j^{r}_{p}\sigma.

[edit] Jet manifolds

The r^{th}\, jet manifold of \pi\, is the set

\{j^{r}_{p}\sigma:p \in \mathcal{M}, \sigma \in \Gamma(\pi)\}

and is denoted J^{r}\pi\,. We may define projections \pi_{r}\, and \pi_{r,0}\, called the source and target projections respectively, by

\pi_{r}:J^{r}\pi\, \longrightarrow \mathcal{M}
j^{r}_{p}\sigma \longmapsto p
\pi_{r,0}:J^{r}\pi\, \longrightarrow \mathcal{E}
j^{r}_{p}\sigma \longmapsto \sigma(p)

If 1 \leq k \leq r, then the k-jet projection is the function \pi_{r,k}\, defined by

\pi_{r,k}:J^{r}\pi \, \longrightarrow J^{k}\pi
j^{r}_{p}\sigma \longmapsto j^{k}_{p}\sigma

From this definition, it is clear that \pi_{r} = \pi \circ \pi_{r,0} and that if 0 \leq m \leq k, then \pi_{r,m} = \pi_{k,m} \circ \pi_{r,k}. It is conventional to regard \pi_{r,r}=id_{J^{r}\pi}\,, the identity map on J^{r}\pi \, and to identify J^{0}\pi\, with \mathcal{E}.

The functions \pi_{r,k}, \pi_{r,0}\, and \pi_{r}\, are smooth surjective submersions.

A co-ordinate system on \mathcal{E} will generate a co-ordinate system on J^{r}\pi\,. Let (U,u)\, be an adapted co-ordinate chart on \mathcal{E}, where u = (x^{i}, u^{\alpha})\,. The induced co-ordinate chart (U^{r}, u^{r})\, on J^{r}\pi\, is defined by

U^{r} \, = \{ j^{r}_{p}\sigma: \sigma(p) \in U \} \,
u^{r} \, = (x^{i}, u^{\alpha}, u^{\alpha}_{I})\,

where

x^{i}(j^{r}_{p}\sigma) \, = x^{i}(p) \,
u^{\alpha}(j^{r}_{p}\sigma) \, = u^{\alpha}(\sigma(p)) \,

and the n \left( {}^{m+r}C_{r} -1\right)\, functions

u^{\alpha}_{I}:U^{k} \longrightarrow \mathbb{R}\,

are specified by

u^{\alpha}_{I}(j^{r}_{p}\sigma) = \left.\frac{\partial^{|I|} \sigma^{\alpha}}{\partial x^{|I|}}\right|_{p}

and are known as the derivative co-ordinates.

Given an atlas of adapted charts (U,u)\, on \mathcal{E}, the corresponding collection of charts (U^{r},u^{r})\, is a finite-dimensional C^{\infty}\, atlas on J^{r}\pi\,.

[edit] Jet bundles

Since the atlas on each J^{r}\pi\, defines a manifold, the triples (J^{r}\pi, \pi_{r,k}, J^{k}\pi), (J^{r}\pi, \pi_{r,0}, \mathcal{E})\, and (J^{r}\pi, \pi_{r}, \mathcal{M})\, all define fibered manifolds. In particular, if (\mathcal{E}, \pi, \mathcal{M})\, is a fiber bundle, the triple (J^{r}\pi, \pi_{r}, \mathcal{M})\, defines the r^{th}\, jet bundle of \pi\,.

If W \subset \mathcal{M}\, is an open submanifold, then

J^{r}\left(\pi|_{\pi^{-1}(W)}\right) \cong \pi^{-1}_{r}(W)\,

If p \in \mathcal{M}\,, then the fiber \pi^{-1}_{r}(p)\, is denoted J^{r}_{p}\pi\,.

Let \sigma\, be a local section of \pi\, with domain W \subset \mathcal{M}\,. The r^{th}\, jet prolongation of \sigma\, is the map j^{r}\sigma:W \longrightarrow J^{r}\pi\, defined by

(j^{r}\sigma)(p) = j^{r}_{p}\sigma \,

Note that \pi_{r} \circ j^{r}\sigma = \operatorname{id}_{W} \,, so j^{r}\sigma\, really is a section. In local co-ordinates, \sigma\, is given by

\left(\sigma^{\alpha}, \frac{\partial^{|I|} \sigma^{\alpha}}{\partial x^{|I|}}\right) \qquad 1 \leq |I| \leq r \,

We identify j^{0}\sigma\, with \sigma\,.

[edit] Example

If \pi\, is the trivial bundle (\mathcal{M} \times \mathbb{R}, pr_{1}, \mathcal{M}), then there is a canonical diffeomorphism between the first jet bundle J^{1}\pi\, and T^{*}\mathcal{M} \times \mathbb{R}. To construct this diffeomorphism, for each \sigma \in \Gamma_{W}(\pi)\, write \bar{\sigma} = pr_{2} \circ \sigma \in C^{\infty}(W)\,.

Then, whenever p \in W \,

j^{1}_{p}\sigma = \{ \psi : \psi \in \Gamma_{p}(\pi); \bar{\psi}(p) = \bar{\sigma}(p); d\bar{\psi}_{p} = d\bar{\sigma}_{p} \} \,

Consequently, the mapping

J^{1}\pi \, \longrightarrow T^{*}\mathcal{M} \times \mathbb{R}
j^{1}_{p}\sigma \, \longmapsto (d\bar{\sigma}_{p},\bar{\sigma}(p)) \,

is well-defined and is clearly injective. Writing it out in co-ordinates shows that it is a diffeomorphism, because if (x^{i},u)\, are co-ordinates on \mathcal{M} \times \mathbb{R}, where u=id_{\mathbb{R}}\, is the identity co-ordinate, then the derivative co-ordinates u_{i}\, on J^{1}\pi\, correspond to the co-ordinates \partial_{i}\, on T^{*}\mathcal{M}\,.

Likewise, if \pi\, is the trivial bundle (\mathbb{R} \times \mathcal{M}, pr_{1}, \mathbb{R}), then there exists a canonical diffeomorphism between J^{1}\pi\, and \mathbb{R} \times T\mathcal{M}\,

[edit] Contact forms

A differential 1-form \theta\, on the space J^{r}\pi\, is called a contact form (ie. \theta \in \Lambda_{C}^{r}\pi\,) if it is pulled back to the zero form on \mathcal{M}\, by all prolongations. In other words, if \theta \in \Lambda^{1}J^{r+1}\pi\,, then \theta \in \Lambda_{C}^{1}\pi_{r+1,r}\, if and only if, for every open submanifold W \subset \mathcal{M}\, and every \sigma \in \Gamma_{W}(\pi)\,,

(j^{k+1}\sigma)^{*}\theta = 0\,

The distribution on J^{r}\pi\, generated by the contact forms is called the Cartan distribution. It is the main geometrical structure on jet spaces and plays an important role in the geometric theory of partial differential equations. The Cartan distributions are not involutive and are of growing dimension when passing to higher order jet spaces. Surprisingly though, when passing to the space of infinite order jets J^\infty this distribution is involutive and finite dimensional. Its dimension coinciding with the dimension of the base manifold \mathcal{M}.

[edit] Example

Let us consider the case (\mathcal{E},\pi,\mathcal{M}), where \mathcal{E} \simeq \mathbb{R}^{2} and \mathcal{M} \simeq \mathbb{R}. Then, (J^{1}\pi, \pi, \mathcal{E}) defines the first jet bundle, and may be co-ordinated by (x,u,u_{1})\,, where

x(j^{1}_{p}\sigma) = x(p) = x\,
u(j^{1}_{p}\sigma) = u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \,
u_{1}(j^{1}_{p}\sigma) = \left.\frac{\partial \sigma}{\partial x}\right|_{p} = \sigma'(x)

for all p \in \mathcal{M} and \sigma \in \Gamma_{p}(\pi)\,. A general 1-form on J^{1}\pi\, takes the form

\theta = a(x, u, u_{1})dx + b(x, u, u_{1})du + c(x, u,u_{1})du_{1}\,

A section \sigma \in \Gamma_{p}(\pi)\, has first prolongation j^{1}\sigma = (u,u_{1}) = \left(\sigma(p), \left.\frac{\partial \sigma}{\partial x}\right|_{p}\right)\,. Hence, (j^{1}\sigma)^{*} \theta\, can be calculated as

(j^{1}_{p}\sigma)^{*} \theta \, = \theta \circ j^{1}_{p}\sigma \,
= a(x, \sigma(x), \sigma'(x))dx + b(x, \sigma(x), \sigma'(x))d(\sigma(x)) + c(x, \sigma(x),\sigma'(x))d(\sigma'(x)) \,
= a(x, \sigma(x), \sigma'(x))dx + b(x, \sigma(x), \sigma'(x))\sigma'(x)dx + c(x, \sigma(x),\sigma'(x))\sigma''(x)dx \,
= [\, a(x, \sigma(x), \sigma'(x)) + b(x, \sigma(x), \sigma'(x))\sigma'(x) + c(x, \sigma(x),\sigma'(x))\sigma''(x)\, ]dx \,

This will vanish for all sections \sigma\, if and only if c=0\, and a = -b\sigma'(x)\,. Hence, \theta=b(x, \sigma(x), \sigma'(x))\theta_{0}\, must necessarily be a multiple of the basic contact form \theta_{0}=du-u_{1}dx\,. Proceeding to the second jet space J^{2}\pi\, with additional co-ordinate u_{2}\,, such that

u_{2}(j^{2}_{p}\sigma)=\left.\frac{\partial^{2} \sigma}{\partial x^{2}}\right|_{p} = \sigma''(x)\,

a general 1-form has the construction

\theta = a(x, u, u_{1},u_{2})dx + b(x, u, u_{1},u_{2})du + c(x, u, u_{1},u_{2})du_{1} + e(x, u, u_{1},u_{2})du_{2}\,

This is a contact form if and only if

(j^{2}_{p}\sigma)^{*} \theta \, = \theta \circ j^{2}_{p}\sigma \,
= a(x, \sigma(x), \sigma'(x),\sigma''(x))dx + b(x, \sigma(x),\sigma'(x),\sigma''(x))d(\sigma(x))+ \,
+ c(x, \sigma(x),\sigma'(x),\sigma'(x))d(\sigma'(x)) + e(x, \sigma(x), \sigma'(x),\sigma''(x))d(\sigma''(x)) \,
= adx + b\sigma'(x)dx + c\sigma''(x)dx + e\sigma'''(x)dx\,
= [\, a + b\sigma'(x) + c\sigma''(x) + e\sigma'''(x)\,]dx\,
= 0\,

which implies that e=0\, and a=-b\sigma'(x)-c\sigma''(x)\,. Therefore, \theta\, is a contact form if and only if

\theta = b(x, \sigma(x), \sigma'(x))\theta_{0} + c(x, \sigma(x), \sigma'(x))\theta_{1}\,

where \theta_{1} = du_{1} - u_{2}dx\, is the next basic contact form (Note that here we are identifying the form \theta_{0}\, with its pull-back (\pi_{2,1})^{*}\theta_{0}\, to J^{2}\pi\,).

In general, providing x,u, \in \mathbb{R}\,, a contact form on J^{r+1}\pi\, can be written as a linear combination of the basic contact forms

\theta_{k} = du_{k} - u_{k+1}dx \qquad k=0, \ldots, r-1\,

where u_{k}(j^{k}\sigma)= \left.\frac{\partial^{k} \sigma}{\partial x^{k}}\right|_{p}\,.

Similar arguments lead to a complete characterization of all contact forms.

In local coordinates, every contact one-form on J^{r+1}\pi\, can be written as a linear combination

\theta = \sum_{|I|=0}^{r} P_{\alpha}^{I}\theta_{I}^{\alpha}\,

with smooth coefficients P^{\alpha}_{I}(x^{i},u^{\alpha})\, of the basic contact forms

\theta_{I}^{\alpha} = du^{\alpha}_{I} - u^{\alpha}_{I+1,i}dx^{i}\,

|I|\, is known as the order of the contact form \theta_{I}^{\alpha}. Note that contact forms on J^{r+1}\pi\, have orders at most r\,. Contact forms provide a characterization of those local sections of \pi_{r+1}\, which are prolongations of sections of \pi\,.

Let \psi \in \Gamma_{W}(\pi_{r+1})\,, then \psi = j^{r+1}\sigma\, where \sigma \in \Gamma_{W}(\pi)\, if and only if \psi^{*}(\sigma|_{W})=0, \forall \sigma \in \Lambda_{C}^{1}\pi_{r+1,r}\,

[edit] Vector fields

A general vector field on the total space \mathcal{E}, co-ordinated by (x,u) \ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha})\,, is

V \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u)\frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u)\frac{\partial}{\partial u^{\alpha}}\,

A vector field is called horizontal, meaning all the vertical coefficients vanish, if \phi^{\alpha}=0\,.

A vector field is called vertical, meaning all the horizontal coefficients vanish, if \rho^{i}=0\,.

For fixed (x,u)\,, we identify

V_{(xu)} \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u) \frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u) \frac{\partial}{\partial u^{\alpha}}\,

having co-ordinates (x,u,\rho^{i},\phi^{\alpha})\,, with an element in the fiber T_{xu}\mathcal{E} of T\mathcal{E} over (x,u) \in \mathcal{E}, called a tangent vector in T\mathcal{E}. A section

\psi : \mathcal{E} \, \longrightarrow T\mathcal{E}
(x,u) \, \longmapsto \psi(x,u) = V\,

is called a vector field on \mathcal{E} with V = \rho^{i}(x,u) \frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u) \frac{\partial}{\partial u^{\alpha}}\, and \psi \in \Gamma(T\mathcal{E})\,.

The jet bundle J^{r}\pi\, is co-ordinated by (x,u,w) \ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha},u_{i}^{\alpha})\,. For fixed (x,u,w)\,, identify

V_{(xuw)} \ \stackrel{\mathrm{def}}{=}\ \, V^{i}(x,u,w) \frac{\partial}{\partial x^{i}} + V^{\alpha}(x,u,w) \frac{\partial}{\partial u^{\alpha}} \ + \ V^{\alpha}_{i}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i}} +\,
\qquad + \ V^{\alpha}_{i_{1}i_{2}}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i_{1}i_{2}}} + \cdots \ + \ \cdots + V^{\alpha}_{i_{1}i_{2} \cdots i_{r}}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i_{1}i_{2} \cdots i_{r}}}\,

having co-ordinates (x,u,w,v^{\alpha}_{i}, v^{\alpha}_{i_{1} i_{2}},\ldots,v^{ \alpha}_{i_{1}i_{2} \cdots i_{r}})\,, with an element in the fiber T_{xuw}(J^{r}\pi)\, of T(J^{r}\pi)\, over (x,u,w) \in J^{r}\pi\,, called a tangent vector in T(J^{r}\pi)\,. Here, v^{\alpha}_{i}, v^{\alpha}_{i_{1}i_{2}},\ldots,v^{\alpha}_{i_{1}i_{2} \cdots i_{r}}\, are real-valued functions on J^{r}\pi\,. A section

\Psi : J^{r}\pi \, \longrightarrow T(J^{r}\pi) \,
(x,u,w) \, \longmapsto \Psi(u,w) = V \,

is a vector field on J^{r}\pi\,, and we say \Psi \in \Gamma(T(J^{r}\pi))\,.

[edit] Partial differential equations

Let (\mathcal{E},\pi,\mathcal{M}) be a fiber bundle. An r^{th}\, order partial differential equation on \pi\, is a closed embedded submanifold \mathcal{S} of the jet manifold J^{r}\pi\,. A solution is a local section \sigma \in \Gamma_{W}(\pi)\, satisfying j^{r}_{p}\sigma \in \mathcal{S}, \forall p \in \mathcal{M}.

Let us consider an example of a first order partial differential equation.

[edit] Example

Let \pi\, be the trivial bundle (\mathbb{R}^{2} \times \mathbb{R}, pr_{1}, \mathbb{R}^{2})\, with global co-ordinates (x^{1}, x^{2}, u^{1})\,. Then the map F:J^{1}\pi \longrightarrow \mathbb{R}\, defined by

F = u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1}\,

gives rise to the differential equation

S = \{ j^{1}_{p}\sigma \in J^{1}\pi : (u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1})(j^{1}_{p}\sigma)=0 \,

which can be written

\frac{\partial \sigma}{\partial x^{1}}\frac{\partial \sigma}{\partial x^{2}} - 2x^{2}\sigma = 0 \,

The particular section \sigma:\mathbb{R}^{2} \longrightarrow \mathbb{R}^{2} \times \mathbb{R}\, defined by

\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2}) \,

has first prolongation given by

j^{1}\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2},(p^{2})^{2},2p^{1}p^{2}) \,

and is a solution of this differential equation, because

(u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1})(j^{1}_{p}\sigma) \, = u^{1}_{1}(j^{1}_{p}\sigma)u^{1}_{2}(j^{1}_{p}\sigma) - 2x^{2}(j^{1}_{p}\sigma)u^{1}(j^{1}_{p}\sigma) \,
= (p^{2})^{2} \cdot 2p^{1}p^{2} - 2 \cdot p^{2} \cdot p^{1}(p^{2})^{2} \,
= 2p^{1}(p^{2})^3 - 2p^{1}(p^{2})^3 \,
= 0 \,

and so j^{1}_{p}\sigma \in \mathcal{S}\, for every p \in \mathbb{R}^{2}\,.

[edit] Jet Prolongation

A local diffeomorphism \psi:J^{r}\pi \longrightarrow J^{r}\pi\, defines a contact transformation of order r\, if it preserves the contact ideal, meaning that if \theta\, is any contact form on J^{r}\pi\,, then j^{r}\psi^{*}\theta\, is also a contact form.

The flow generated by a vector field V^{r}\, on the jet space J^{r}\, forms a one-parameter group of contact transformations if and only if the Lie derivative \mathcal{L}_{V^{r}}(\theta) of any contact form \theta\, preserves the contact ideal.

Let us begin with the first order case. Consider a general vector field V^{1}\, on J^{1}\pi\,, given by

V^{1} \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(u^{1})\frac{\partial}{\partial x^{i}} + \phi^{\alpha}(u^{1})\frac{\partial}{\partial u^{\alpha}} + \chi^{\alpha}_{i}(u^{1})\frac{\partial}{\partial u^{\alpha}_{i}} \,

We now apply \mathcal{L}_{V^{1}} to the basic contact forms \theta^{\alpha} = du^{\alpha} - u_{i}^{\alpha}dx^{i}\,, and obtain

\mathcal{L}_{V^{1}}(\theta^{\alpha}) = \mathcal{L}_{V^{1}}(du^{\alpha} - u_{i}^{\alpha}dx^{i})
= \mathcal{L}_{V^{1}}du^{\alpha} - (\mathcal{L}_{V^{1}}y_{i}^{\alpha})dx^{i} - u_{i}^{\alpha}(\mathcal{L}_{V^{1}}dx^{i}) \,
= d(V^{1}u^{\alpha}) - V^{1}u_{i}^{\alpha}dx^{i} - u_{i}^{\alpha}d(V^{1}x^{i}) \,
= d\phi^{\alpha} - \chi^{\alpha}_{i}dx^{i} - u_{i}^{\alpha}d\rho^{i} \,
= \frac{\partial \phi^{\alpha}}{\partial x^{i}}\, dx^{i} + \frac{\partial \phi^{\alpha}}{\partial u^{k}}\, du^{k} + \frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}}\, du^{k}_{i} - \chi^{\alpha}_{i}dx^{i} - u_{i}^{\alpha}\left[ \frac{\partial \rho^{i}}{\partial x^{m}}\, dx^{m} + \frac{\partial \rho^{i}}{\partial u^{k}}\, du^{k} + \frac{\partial \rho^{i}}{\partial u^{k}_{m}}\, dy^{k}_{m} \right ] \,

where we have expanded the exterior derivative of the functions in terms of their co-ordinates. Next, we note that

\theta^{k} = du^{k} - u_{i}^{k}dx^{i} \quad \Longrightarrow \quad du^{k} = \theta^{k} + u_{i}^{k}dx^{i} \,

and so we may write

\mathcal{L}_{V^{1}}(\theta^{\alpha}) \, = \frac{\partial \phi^{\alpha}}{\partial x^{i}}\, dx^{i} + \frac{\partial \phi^{\alpha}}{\partial u^{k}}\, (\theta^{k} + u_{i}^{k}dx^{i}) + \frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}}\, du^{k}_{i} - \chi^{\alpha}_{i}dx^{i} - \,
- u_{l}^{\alpha}\left[ \frac{\partial \rho^{l}}{\partial x^{i}}\, dx^{i} + \frac{\partial \rho^{l}}{\partial u^{k}}\, (\theta^{k} + u_{i}^{k}dx^{i}) + \frac{\partial \rho^{l}}{\partial u^{k}_{i}}\, dy^{k}_{i} \right ] \,
= \left[ \frac{\partial \phi^{\alpha}}{\partial x^{i}} + \frac{\partial \phi^{\alpha}}{\partial u^{k}}u_{i}^{k} - y_{l}^{\alpha}\left(\frac{\partial \rho^{l}}{\partial x^{i}} + \frac{\partial \rho^{l}}{\partial u^{k}}u_{i}^{k}\right)- \chi^{\alpha}_{i}\right]\, dx^{i} + \left[ \frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}} - y_{l}^{\alpha}\frac{\partial \rho^{l}}{\partial u^{k}_{i}}\right]\, du^{k}_{i} + \,
+ \left( \frac{\partial \phi^{\alpha}}{\partial u^{k}} - u_{l}^{\alpha}\frac{\partial \rho^{l}}{\partial u^{k}} \right)\theta^{k}\,

Therefore, V^{1}\, determines a contact transformation if and only if the coefficients of dx^{i}\, and dy^{k}_{i}\, in the formula vanish. The latter requirements imply the contact conditions

\frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}} - u^{\alpha}_{l} \frac{\partial \rho^{l}}{\partial u^{k}_{i}} = 0\,

The former requirements provide explicit formulae for the coefficients of the first derivative terms in V^{1}\,:

\chi^{\alpha}_{i} = \widehat{D}_{i} \phi^{\alpha} - u^{\alpha}_{l}(\widehat{D}_{i}\rho^{l}) where \widehat{D}_{i} = \frac{\partial}{\partial x^{i}} + u^{k}_{i}\frac{\partial}{\partial u^{k}}

denotes the zeroth order truncation of the total derivative D_{i}\,.

Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if \mathcal{L}_{V^{r}}\, satisfies these equations, V^{r}\, is called the r^{th}\, prolongation of V\, to a vector field on J^{r}\pi\,.

These results are best understood when applied to a particular example. Hence, let us examine the following.

[edit] Example

Let us consider the case (\mathcal{E},\pi,\mathcal{M}), where \mathcal{E} \simeq \mathbb{R}^{2} and \mathcal{M} \simeq \mathbb{R}. Then, (J^{1}\pi, \pi, \mathcal{E}) defines the first jet bundle, and may be co-ordinated by (x,u,u_{1})\,, where

x(j^{1}_{p}\sigma) \, = x(p) = x \,
u(j^{1}_{p}\sigma) \, = u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \,
u_{1}(j^{1}_{p}\sigma) \, = \left.\frac{\partial \sigma}{\partial x}\right|_{p} = \dot{\sigma}(x) \,

for all p \in \mathcal{M} and \sigma \in \Gamma_{p}(\pi)\,. A contact form on J^{1}\pi\, has the form

\theta = du - u_{1}dx \,

Let us consider a vector V\, on \mathcal{E}, having the form

V = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} \,

Then, the first prolongation of this vector field to J^{1}\pi\, is

V^{1} \, = V + Z \,
= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + Z \,
= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + \rho(x,u,u_{1})\frac{\partial}{\partial u_{1}} \,

If we now take the Lie derivative of the contact form with respect to this prolonged vector field, \mathcal{L}_{V^{1}}(\theta)\,, we obtain

\mathcal{L}_{V^{1}}(\theta) \, = \mathcal{L}_{V^{1}}(du - u_{1}dx) \,
= \mathcal{L}_{V^{1}}du - (\mathcal{L}_{V^{1}}u_{1})dx - u_{1}(\mathcal{L}_{V^{1}}dx) \,
= d(V^{1}u) - V^{1}u_{1}dx - u_{1}d(V^{1}x) \,
= dx - \rho(x,u,u_{1})dx + u_{1}du \,
= (1 - \rho(x,u,u_{1}) )dx + u_{1}du \,

But, we may identify du = \theta + u_{1}dx\,. Thus, we get

\mathcal{L}_{V^{1}}(\theta) \, = [\,1 - \rho(x,u,u_{1})\,]dx + u_{1}(\theta + u_{1}dx) \,
= [\,1 + u_{1}u_{1} - \rho(x,u,u_{1})\,]dx + u_{1}\theta \,

Hence, for \mathcal{L}_{V^{1}}(\theta)\, to preserve the contact ideal, we require

1 + u_{1}u_{1} - \rho(x,u,u_{1}) = 0 \,
\Longrightarrow \quad \, \rho(x,u,u_{1}) = 1 + u_{1}u_{1}\,

And so the first prolongation of V\, to a vector field on J^{1}\pi\, is

V^{1} = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + (1 + u_{1}u_{1})\frac{\partial}{\partial u_{1}} \,

Let us also calculate the second prolongation of V\, to a vector field on J^{2}\pi\,. We have \{x,u,u_{1}, y_{2}\}\, as co-ordinates on J^{2}\pi\,. Hence, the prolonged vector has the form

V^{2} = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + \rho(x,u,u_{1},u_{2})\frac{\partial}{\partial u_{1}} + \phi(x,u,u_{1},u_{2})\frac{\partial}{\partial u_{2}} \,

The contacts forms are

\theta \, = du - u_{1}dx \,
\theta_{1} \, = du_{1} - u_{2}dx \,

To preserve the contact ideal, we require

\mathcal{L}_{V^{2}}(\theta) \, = 0\,
\mathcal{L}_{V^{2}}(\theta_{1}) \, = 0 \,

Now, \theta\, has no u_{2}\, dependency. Hence, from this equation we will pick up the formula for \rho\,, which will necessarily be the same result as we found for V^{1}\,. Therefore, the problem is analogous to prolonging the vector field V^{1}\, to J^{2}\pi\,. That is to say, we may generate the r^{th}\,-prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, r\, times. So, we have

\rho(x,u,u_{1}) = 1 + u_{1}u_{1} \,

and so

V^{2} \, = V^{1} + \phi(x,u,u_{1},u_{2})\frac{\partial}{\partial u_{2}} \,
= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + (1 + u_{1}u_{1})\frac{\partial}{\partial u_{1}} + \phi(x,u,u_{1},u_{2})\frac{\partial}{\partial u_{2}} \,

Therefore, the Lie derivative of the second contact form with respect to V^{2}\, is

\mathcal{L}_{V^{2}}(\theta_{1}) \, = \mathcal{L}_{V^{2}}(du_{1} - u_{2}dx) \,
= \mathcal{L}_{V^{2}}du_{1} - (\mathcal{L}_{V^{2}}u_{2})dx - u_{2}(\mathcal{L}_{V^{2}}dx) \,
= d(V^{2}u_{1}) - V^{2}u_{2}dx - u_{2}d(V^{2}x) \,
= d(1-u_{1}u_{1}) - \phi(x,u,u_{1},u_{2})dx + u_{2}du \,
= 2u_{1}du_{1} - \phi(x,u,u_{1},u_{2})dx + u_{2}du \,

Again, let us identify du=\theta + u_{1}dx \, and du_{1}=\theta_{1} + u_{2}dx \,. Then we have

\mathcal{L}_{V^{2}}(\theta_{1}) \, = 2u_{1}(\theta_{1} + u_{2}dx) - \phi(x,u,u_{1},u_{2})dx + u_{2}(\theta + u_{1}dx) \,
= [\, 3u_{1}u_{2} - \phi(x,u,u_{1},u_{2})\,]dx + u_{2}\theta + 2u_{1}\theta_{1} \,

Hence, for \mathcal{L}_{V^{2}}(\theta_{1})\, to preserve the contact ideal, we require

3u_{1}u_{2} - \phi(x,u,u_{1},u_{2}) = 0 \,
\Longrightarrow \quad \, \phi(x,u,u_{1},u_{2}) = 3u_{1}u_{2} \,

And so the second prolongation of V\, to a vector field on J^{2}\pi\, is

V^{2} = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + (1 + u_{1}u_{1})\frac{\partial}{\partial u_{1}} + 3u_{1}u_{2}\frac{\partial}{\partial u_{2}} \,

Note that the first prolongation of V\, can be recovered by omitting the second derivative terms in V^{2}\,, or by projecting back to J^{1}\pi\,.

[edit] Infinite Jet Spaces

The inverse limit of the sequence of projections \pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi) gives rise to the infinite jet space J^\infty(\pi). A point j_p^\infty(\sigma) is the equivalence class of sections of π that have the same k-jet in p as σ for all values of k. The natural projection \pi_\infty maps j_p^\infty(\sigma) into p.

Just by thinking in terms of coordinates, J^\infty(\pi) appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on J^\infty(\pi), not relying on differentiable charts, is given by the differential calculus over commutative algebras. Dual to the sequence of projections \pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi) of manifolds is the sequence of injections \pi_{k+1,k}^*:C^\infty(J^{k}(\pi))\to C^\infty(J^{k+1}(\pi)) of commutative algebras. Let's denote C^\infty(J^{k}(\pi)) simply by \mathcal{F}_k(\pi). Take now the direct limit \mathcal{F}(\pi) of the \mathcal{F}_k(\pi)'s. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object J^\infty(\pi). Observe that \mathcal{F}(\pi), being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

Roughly speaking, a concrete element \varphi\in\mathcal{F}(\pi) will always belong to some \mathcal{F}_k(\pi), so it is a smooth function on the finite-dimensional manifold Jk(π) in the usual sense.

[edit] Infinitely prolonged PDE's

Given a k-th order system of PDE's \mathcal{E}\subseteq J^k(\pi), the collection I(\mathcal{E}) of vanishing on \mathcal{E} smooth functions on J^\infty(\pi) is an ideal in the algebra \mathcal{F}_k(\pi), and hence in the direct limit \mathcal{F}(\pi) too.

Enhance I(\mathcal{E}) by adding all the possible compositions of total derivatives applied to all its elements. This way we get a new ideal I of \mathcal{F}(\pi) which is now closed under the operation of taking total derivative. The submanifold \mathcal{E}_{(\infty)} of J^\infty(\pi) cut out by I is called the infinite prolongation of \mathcal{E}.

Geometrically, \mathcal{E}_{(\infty)} is the manifold of formal solutions of \mathcal{E}. A point j_p^\infty(\sigma) of \mathcal{E}_{(\infty)} can be easily seen to be represented by a section σ whose k-jet's graph is tangent to \mathcal{E} at the point j_p^k(\sigma) with arbitrarily high order of tangency.

Analitically, if \mathcal{E} is given by \varphi=0, a formal solution can be understood as the set of Taylor coefficients of a section σ in a point p that make vanish the Taylor series of \varphi\circ j^k(\sigma) at the point p.

Most importantly, the closure properties of I imply that \mathcal{E}_{(\infty)} is tangent to the infinite-order contact structure \mathcal{C} on J^\infty(\pi), so that by restricting \mathcal{C} to \mathcal{E}_{(\infty)} one gets the diffiety (\mathcal{E}_{(\infty)},\mathcal{C}|_{\mathcal{E}_{(\infty)}}), and can study the associated C-spectral sequence.

[edit] Remark

This article has defined jets of local sections of a bundle, but it is possible to define jets of functions f:\mathcal{M} \longrightarrow \mathcal{N}\,, where \mathcal{M} and \mathcal{N} are manifolds; the jet of f\, then just corresponds to the jet of the section

gr_{f}:\mathcal{M} \, \longrightarrow \mathcal{M} \times \mathcal{N} \,
p \, \longmapsto gr_{f}(p) = (p, f(p) )\,

(gr_{f}\, is known as the graph of the function f\,) of the trivial bundle (\mathcal{M} \times \mathcal{N}, \pi_{1}, \mathcal{F}). However, this restriction does not simplify the theory, as the global triviality of \pi\, does not imply the global triviality of \pi_{1}\,.

[edit] References

  • Ehresmann, C., "Introduction a la théorie des structures infinitésimales et des pseudo-groupes de mathcal{L}." Geometrie Differentielle, Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127.
  • Kolár, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. ISBN 3-540-56235-4, ISBN 0-387-56235-4.
  • Saunders, D.J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
  • Krasil'shchik, I.S., Vinogradov, A.M., [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958-X.
  • Olver, P.J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0-521-47811-1
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